Segmenting a model within a plasma system

ABSTRACT

Systems and methods for segmenting an impedance matching model are described. One of the methods includes receiving the impedance matching model. The impedance matching model represents an impedance matching circuit, which is coupled to an RF generator via an RF cable and to a plasma chamber via an RF transmission line. The method further includes segmenting the impedance matching model into two or more modules of a first set. Each module includes a series circuit and a shunt circuit. The shunt circuit is coupled to the series circuit. The series circuit of the first module is coupled to a cable model and the series circuit of the second module is coupled to an RF transmission model. The series circuit and the shunt circuit of the first module are coupled to the series circuit of the second module. The shunt circuit of the second module is coupled to the RF transmission model.

CLAIM OF PRIORITY

This application is a continuation of and claims the benefit of andpriority, under 35 U.S.C. § 120, to U.S. patent application Ser. No.14/245,803, filed on Apr. 4, 2014, and titled “Segmenting a Model Withina Plasma System”, which claims the benefit of and priority, under 35U.S.C. § 119(e), to U.S. Provisional Patent Application No. 61/821,523,filed on May 9, 2013, and titled “Segmenting a Model Within a PlasmaSystem”, both of which are hereby incorporated by reference in theirentirety

This U.S. patent application Ser. No. 14/245,803 is acontinuation-in-part (CIP) of and claims the benefit of and priority,under 35 U.S.C. § 120, to U.S. patent application Ser. No. 13/756,390,filed on Jan. 31, 2013, and titled “Using Modeling to Determine WaferBias Associated with a Plasma System”, now issued as U.S. Pat. No.9,502,216, which is hereby incorporated by reference in its entirety.

FIELD

The present embodiments relate to generating segments of a model withina plasma system.

BACKGROUND

A plasma-based system is used to perform a variety of operations. Forexample, the plasma-based system is used to etch a wafer, depositmaterials on a wafer, clean a wafer, etc. To perform the operations, theplasma-based system includes a radio frequency (RF) generator. The RFgenerator is coupled to an impedance block that is further coupled to aplasma chamber.

The RF generator generates an RF signal that is transferred via theimpedance block to the plasma chamber. When a gas is supplied into theplasma chamber, the gas is ignited with the RF signal and plasma isformed within the plasma chamber.

However, there may be a replacement of the impedance block with anotherimpedance block. For example, an impedance block that is malfunctioningmay be replaced with another impedance block. As another example, animpedance block that is nonoperational may be replaced with anotherimpedance block. An impedance block may be replaced for any reason otherthan that the impedance block is nonoperational or malfunctioning.

It is in this context that embodiments described in the presentdisclosure arise.

SUMMARY

Embodiments of the disclosure provide apparatus, methods and computerprograms for generating segments of a model within a plasma system. Itshould be appreciated that the present embodiments can be implemented innumerous ways, e.g., a process, an apparatus, a system, a piece ofhardware, or a method on a computer-readable medium. Several embodimentsare described below.

In various embodiments, a model is formed from a circuit of a plasmasystem. For example, an impedance matching model is formed based oncharacteristics of an impedance matching circuit, a cable model isformed based on characteristics of a radio frequency (RF) cable, or anRF transmission model is formed based on characteristics of an RFtransmission line. The model is segmented into a number of modules. Eachmodule includes a series circuit and a shunt circuit. When a circuit ofthe plasma system is to be replaced with another circuit of the plasmasystem, one or more of the modules is easily replaced with one or moremodules.

In various embodiments, a method for segmenting an impedance matchingmodel is described. The method includes receiving the impedance matchingmodel. The impedance matching model represents an impedance matchingcircuit, which is coupled to an RF generator via an RF cable and to aplasma chamber via an RF transmission line. The method further includessegmenting the impedance matching model into two or more modules of afirst set. Each module including a series circuit and a shunt circuit.The shunt circuit is coupled to the series circuit. The shunt circuit iscoupled to a ground connection. The series circuit of the first moduleis coupled to a cable model. The series circuit of the second module iscoupled to an RF transmission model. The series circuit of the firstmodule is coupled to the series circuit of the second module. The shuntcircuit of the first module coupled to the series circuit of the secondmodule. The shunt circuit of the second module is coupled to the RFtransmission model. The method is executed by a processor.

In some embodiments, a method for segmenting an RF transmission model isdescribed. The method includes receiving an RF transmission model, whichrepresents an RF transmission line. The RF transmission line couples aplasma chamber to an impedance matching circuit, which is coupled via anRF cable to an RF generator. The method further includes segmenting theRF transmission model into two or more modules of a first set. Eachmodule includes a series circuit and a shunt circuit. The shunt circuitis coupled to the series circuit. The shunt circuit is coupled to aground connection and the series circuit of the first module is coupledto an impedance matching model. The series circuit of the first modulecoupled to the series circuit of the second module and the shunt circuitof the first module is coupled to the series circuit of the secondmodule. The method is executed by a processor.

In a variety of embodiments, a method for segmenting a cable model isdescribed. The method includes receiving a cable model, the cable modelrepresenting an RF cable, which couples an RF generator to an impedancematching circuit. The impedance matching circuit is coupled via an RFtransmission line to a plasma chamber. The method includes segmentingthe cable model into two or more modules of a first set. Each moduleincludes a series circuit and a shunt circuit. The shunt circuit iscoupled to the series circuit and to a ground connection. The seriescircuit of the first module receives a complex voltage and current froma voltage and current probe. The shunt circuit of the second module iscoupled to an impedance matching model. The series circuit of the secondmodule is coupled to the impedance matching model. The method isexecuted by a processor.

In various embodiments, a method for segmenting an impedance matchingmodel is described. The method includes receiving the impedance matchingmodel, which represents an impedance matching circuit. The impedancematching circuit is coupled to an RF generator via an RF cable and to aplasma chamber via an RF transmission line. The method further includessegmenting the impedance matching model into two or more modules of afirst set. Each module includes a series circuit and a shunt circuit.The shunt circuit is coupled to the series circuit and to a groundconnection. The series circuit of a first one of the modules is coupledto a cable model and the shunt circuit of the first module is coupled tothe cable model. The series circuit of the first module is coupled tothe series circuit of the second module and the series circuit of thesecond module coupled to the RF transmission model. The shunt circuit ofthe second module is coupled to the series circuit of the first module.The method is executed by a processor.

In some embodiments, a method for segmenting an impedance matching modelis described. The method includes receiving the impedance matchingmodel, which represents an impedance matching circuit. The impedancematching circuit is coupled to an RF generator via an RF cable and to aplasma chamber via an RF transmission line. The method further includessegmenting the impedance matching model into two or more modules. Eachmodule includes a series function and a shunt function. The shuntfunction is coupled to the series function and to a ground function. Theseries function of a first one of the modules is coupled to a cablemodel and the series function of a second one of the modules coupled toan RF transmission model. Also, the series function of the first moduleis coupled to the series function of the second module and the shuntfunction of the first module is coupled to the series function of thesecond module. The shunt function of the second module is coupled to theRF transmission model. The method is executed by a processor.

In several embodiments, a method for segmenting an impedance matchingmodel is described. The method includes receiving the impedance matchingmodel, which represents an impedance matching circuit. The impedancematching circuit is coupled to an RF generator via an RF cable and to aplasma chamber via an RF transmission line. The method further includessegmenting the impedance matching model into two or more modules of afirst set. Each module includes a series function and a shunt function.The shunt function is coupled to the series function and to a groundfunction. The series function of a first one of the modules is coupledto a cable model and the shunt function of the first module is coupledto the cable model. The series function of the first module is coupledto the series function of the second module and the series function ofthe second module is coupled to the RF transmission model. The shuntfunction of the second module is coupled to the series function of thefirst module. The method is executed by a processor.

In one embodiment, a method for segmenting an impedance matching model,the method is described. The method includes generating, by a computer,the impedance matching model. The impedance matching model represents animpedance matching circuit. The impedance matching circuit is configuredto couple to an RF generator via an RF cable and to a plasma chamber viaan RF transmission line. The impedance matching model includes a firstmodule for a portion of the impedance matching circuit. The methodfurther includes replacing the first module with one or more othermodules when the impedance matching circuit is replaced with anotherimpedance matching circuit. The method is executable by a processor.

In an embodiment, the first module includes a series circuit. In oneembodiment, the series circuit includes a combination of a resistor, acapacitor, and an inductor.

In an embodiment, the first module is coupled to a second module. Thesecond module is coupled between the first module and acomputer-generated model of the RF cable. The series circuit has a firstend that is coupled to the second module. The series circuit has asecond end that is coupled to a computer-generated model of the RFtransmission line.

In one embodiment, the first module is coupled to a second module. Thesecond module is located between the first module and acomputer-generated model of the RF transmission line. The series circuithas a first end that is coupled to a computer-generated model of the RFcable and has a second end that is coupled to the second module.

In an embodiment, the first module includes a shunt circuit having afirst end that is coupled to a ground connection. In one embodiment, theshunt circuit includes a combination of a resistor, a capacitor, and aninductor.

In one embodiment, the first module is coupled to a second module. In anembodiment, the second module is coupled between the first module and acomputer-generated model of the RF cable. The shunt circuit has a secondend that is coupled to the second module and to a computer-generatedmodel of the RF transmission line.

In an embodiment, the first module is coupled to a second module. Thesecond module is coupled between the first module and acomputer-generated model of the RF transmission line. The shunt circuithas a second end that is coupled to the second module and to acomputer-generated model of the RF cable.

In one embodiment, the first module is a polynomial function defining aseries circuit. In an embodiment, the polynomial function includes acombination of a resistance and a reactance.

In one embodiment, the first module is a polynomial functionrepresenting a shunt circuit. In an embodiment, the polynomial functionincludes a combination of a resistance and a reactance.

In one embodiment, the one or more other modules represent a portion ofthe other impedance matching circuit.

Some advantages of the above-described embodiments include ease inreplacement of one module of a model with another module of a model. Forexample, when an impedance matching circuit is replaced with anotherimpedance matching circuit, one or more modules of an impedance matchingmodel that represents the impedance matching circuit being replaced iseasily switched with one or more modules of a replacement impedancematching model that represents the replacement impedance matchingcircuit. For example, a computer-generated code for the one or moremodules of the replacement impedance matching model can be easilyreplaced with a computer-generated code of the one or more modules ofthe impedance matching model being replaced. Similarly, as anotherexample, when an RF cable is replaced with another RF cable, one or moremodules of a cable model that represents the RF cable being replaced iseasily switched with one or more modules of another cable model thatrepresents the replacement RF cable. Also, as another example, when anRF transmission line is replaced with another RF transmission line, oneor more modules of an RF transmission model that represents the RFtransmission line being replaced are is switched with one or moremodules of another RF transmission model that represents the replacementRF transmission line.

Other aspects will become apparent from the following detaileddescription, taken in conjunction with the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

The embodiments may best be understood by reference to the followingdescription taken in conjunction with the accompanying drawings.

FIG. 1 is a block diagram of a plasma system for segmenting an impedancematching model, a cable model and/or a radio frequency (RF) transmissionmodel, in accordance with some embodiments of the present disclosure.

FIG. 2 is a diagram of an impedance matching model after conversion ofthe impedance matching model of FIG. 1, in accordance with severalembodiments of the present disclosure.

FIG. 3 is a diagram of a module of the converted impedance matchingmodel, in accordance with various embodiments of the present disclosure.

FIG. 4A is a diagram of the module of FIG. 3 in which inductors andcapacitors have fixed values, in accordance with some embodiments of thepresent disclosure.

FIG. 4B is a diagram of the module of FIG. 3 in which inductors havevariable values, in accordance with several embodiments of the presentdisclosure.

FIG. 4C is a diagram of the module of FIG. 3 in which capacitors havevariable values, in accordance with various embodiments of the presentdisclosure.

FIG. 4D is a diagram of the module of FIG. 3 in which inductors andcapacitors have variable values, in accordance with some embodiments ofthe present disclosure.

FIG. 4E is a diagram of a module that includes a functionalrepresentation of a series circuit of the module of FIG. 3 and afunctional representation of a shunt circuit of the module of FIG. 3, inaccordance with various embodiments of the present disclosure.

FIG. 5A is a diagram of a circuit of the impedance matching model ofFIG. 1, in accordance with various embodiments of the presentdisclosure.

FIG. 5B is a diagram of a segmented circuit generated from the circuitof FIG. 5A, in accordance with some embodiments of the presentdisclosure.

FIG. 5C is a diagram of a segmented circuit that is generated from thesegmented circuit of FIG. 5B, in accordance with several embodiments ofthe present disclosure.

FIG. 6 is a diagram of an embodiment of a module of the impedancematching model to illustrate a change in positions of a shunt circuitand a series circuit of the impedance matching model compared topositions of the shunt and series circuits illustrated in FIG. 3.

FIG. 7A is a diagram of an embodiment of a diagram of the module of FIG.6 in which inductors and capacitors have fixed values, in accordancewith some embodiments of the present disclosure.

FIG. 7B is a diagram of the module of FIG. 6 in which inductors havevariable values, in accordance with several embodiments of the presentdisclosure.

FIG. 7C is a diagram of the module of FIG. 6 in which capacitors havevariable values, in accordance with various embodiments of the presentdisclosure.

FIG. 7D is a diagram of the module of FIG. 6 in which inductors andcapacitors have variable values, in accordance with some embodiments ofthe present disclosure.

FIG. 7E is a diagram of a module that includes a functionalrepresentation of a series circuit of the module of FIG. 6 and afunctional representation of a shunt circuit of the module of FIG. 6, inaccordance with various embodiments of the present disclosure.

FIG. 8 is a diagram of a segmented cable model generated from the cablemodel of FIG. 1 or a segmented RF transmission model generated from theRF transmission model of FIG. 1, in accordance with several embodimentsof the present disclosure.

FIG. 9 is a diagram of a module of the RF cable model/Transmission linemodel of FIG. 8, in accordance with some embodiments of the presentdisclosure.

FIG. 10A is a diagram of the module of FIG. 9 in which an inductance ofan inductor and a capacitance of a capacitor are fixed, in accordancewith various embodiments of the present disclosure.

FIG. 10B is a diagram of a module of FIG. 9 in which an inductance of aninductor is variable, in accordance with several embodiments of thepresent disclosure.

FIG. 10C is a diagram of a module of FIG. 9 in which a capacitance of acapacitor is variable, in accordance with several embodiments of thepresent disclosure.

FIG. 10D is a diagram of a module of FIG. 9 in which an inductance of aninductor and a capacitance of a capacitor are variable, in accordancewith several embodiments of the present disclosure.

FIG. 10E is a diagram of a module that represents a function applied bya series circuit of the module of FIG. 9 and a function that is appliedby a shunt circuit of the module of FIG. 9, in accordance with variousembodiments of the present disclosure.

FIG. 11A is a graph that illustrates a linear relationship between avoltage measured at an output of an impedance matching circuit and amodeled voltage at an output of a corresponding segmented impedancematching model, in accordance with some embodiments of the presentdisclosure.

FIG. 11B is a graph that illustrates a linear relationship between acurrent measured at an output of an impedance matching circuit and amodeled current at an output of a corresponding segmented impedancematching model, in accordance with various embodiments of the presentdisclosure.

FIG. 12A is a graph that illustrates a relationship between a voltagemeasured at an output of an impedance matching circuit with respect totime and a modeled voltage at a corresponding output of an impedancematching model generated based on the impedance matching circuit withrespect to time, in accordance with various embodiments of the presentdisclosure.

FIG. 12B is an embodiment of a graph that illustrates a relationshipbetween a current measured at an output of an impedance matching circuitwith respect to time and a modeled current at a corresponding output ofan impedance matching model generated based on the impedance matchingcircuit with respect to time, in accordance with various embodiments ofthe present disclosure.

DETAILED DESCRIPTION

The following embodiments describe systems and methods for segmenting amodel within a plasma system. It will be apparent that the presentembodiments may be practiced without some or all of these specificdetails. In other instances, well known process operations have not beendescribed in detail in order not to unnecessarily obscure the presentembodiments.

FIG. 1 is a block diagram of an embodiment of a plasma system 100 forsegmenting an impedance matching model 102, a cable model 104A, and/or aradio frequency (RF) transmission model 106. The plasma system 100includes an x megahertz (MHz) RF generator, a y MHz RF generator, and az MHz RF generator.

A voltage and current (VI) probe 108 measures a complex voltage andcurrent Vx, Ix, and φx at an output 110, of the x MHz RF generator. Itshould be noted that Vx represents a voltage magnitude, Ix represents acurrent magnitude, and φx represents a phase between Vx and Ix.Similarly, a voltage and current probe 112 measures a complex voltageand current Vy, Iy, and φy at an output 114 of the y MHz RF generator.It should be noted that Vy represents a voltage magnitude, Iy representsa current magnitude, and φy represents a phase between Vy and Iy.Moreover, a voltage and current probe 116 measures a complex voltage andcurrent Vz, Iz, and φz at an output 118 of the z MHz RF generator. Itshould be noted that Vz represents a voltage magnitude, Iz represents acurrent magnitude, and φz represents a phase between Vz and Iz.

Examples of x MHz include 2 MHz, 27 MHz, and 60 MHz. Examples of y MHzinclude 2 MHz, 27 MHz, and 60 MHz. Examples of z MHz include 2 MHz, 27MHz, and 60 MHz. The x MHz is different than y MHz and z MHz. Forexample, when x MHz is 2 MHz, y MHz is 27 MHz and z MHz is 60 MHz. Whenx MHz is 27 MHz, y MHz is 60 MHz and z MHz is 2 MHz.

An example of a voltage and current probe includes a voltage and currentprobe that complies with a pre-set formula. An example of the pre-setformula includes a standard that is followed by an Association, whichdevelops standards for sensors. Another example of the pre-set formulaincludes a National Institute of Standards and Technology (NIST)standard. As an illustration, the voltage and current probe 108, 112, or116 is calibrated according to NIST standard. In this illustration, thevoltage and current probe 108, 112, or 116 is coupled with an opencircuit, a short circuit, or a known load to calibrate the voltage andcurrent probe to comply with the NIST standard. The voltage and currentprobe 108, 112, or 116 may first be coupled with the open circuit, thenwith the short circuit, and then with the known load to calibrate thevoltage and current probe based on NIST standard. The voltage andcurrent probe 108, 112, or 116 may be coupled to the known load, theopen circuit, and the short circuit in any order to calibrate thevoltage and current probe according to NIST standard. Examples of aknown load include a 50 ohm load, a 100 ohm load, a 200 ohm load, astatic load, a direct current (DC) load, a resistor, etc. As anillustration, each voltage and current probe 108, 112, or 116 iscalibrated according NIST-traceable standards.

The voltage and current probe 108 is coupled to the output 110 of the xMHz RF generator. The output 110 is coupled to an input 120A of animpedance matching circuit 122 via an RF cable 124A. Similarly, thevoltage and current probe 112 is coupled to the output 114 of the y MHzRF generator. The output 114 is coupled to another input 120B of theimpedance matching circuit 122 via an RF cable 124B. Also, the voltageand current probe 116 is coupled to the output 118 of the z MHz RFgenerator. The output 118 is coupled to another input 120C of theimpedance matching circuit 122 via an RF cable 124C.

An output 126 of the impedance matching circuit 122 is coupled to aninput of an RF transmission line 128. The RF transmission line 128 iscoupled to an electrostatic chuck (ESC) 132 located within a plasmachamber 130.

The impedance matching circuit 122 matches an impedance of a sourcecoupled to the impedance matching circuit 122 with an impedance of aload coupled to the impedance matching circuit 122. For example, theimpedance matching circuit 122 matches a combined impedance of the x MHzRF generator and the RF cable 124A with a combined impedance of the RFtransmission line 128 and the plasma chamber 130. In this example, the xMHz RF generator is on and the y and z MHz RF generators are off.

The plasma chamber 130 includes the ESC 132, an upper electrode 134, andother parts (not shown), e.g., an upper dielectric ring surrounding theupper electrode 134, an upper electrode extension surrounding the upperdielectric ring, a lower dielectric ring surrounding a lower electrodeof the ESC 132, a lower electrode extension surrounding the lowerdielectric ring, an upper plasma exclusion zone (PEZ) ring, a lower PEZring, etc. The upper electrode 134 is located opposite to and facing theESC 132. A work piece 136, e.g., a semiconductor wafer, a dummy wafer,etc., is supported on an upper surface 138 of the ESC 132. Variousprocesses, e.g., chemical vapor deposition, cleaning, deposition,sputtering, etching, ion implantation, resist stripping, etc., areperformed on the semiconductor wafer during production. Integratedcircuits, e.g., application specific integrated circuit (ASIC),programmable logic device (PLD), etc. are developed on the semiconductorwafer and the integrated circuits are used in a variety of electronicitems, e.g., cell phones, tablets, smart phones, computers, laptops,networking equipment, etc. Each of the lower electrode and the upperelectrode 134 is made of a metal, e.g., aluminum, alloy of aluminum,copper, etc.

In one embodiment, the upper electrode 134 includes one or more gasinlets, e.g. holes, etc., that is coupled to a central gas feed (notshown). The central gas feed receives one or more process gases from agas supply (not shown). Examples of a process gases include anoxygen-containing gas, such as O₂. Other examples of a process gasinclude a fluorine-containing gas, e.g., tetrafluoromethane (CF₄),sulfur hexafluoride (SF₆), hexafluoroethane (C₂F₆), etc. The upperelectrode 134 is grounded. The ESC 132 is coupled to the x, y, and z MHzRF generators via the impedance matching circuit 122.

When the process gas is supplied between the upper electrode 134 and theESC 132 and when the x MHz RF generator, the y MHz, and/or the z MHz RFgenerator supplies RF signals via the impedance matching circuit 122 andthe RF transmission line 128 to the ESC 132, the process gas is ignitedto generate plasma within the plasma chamber 130.

When the x MHz RF generator generates and provides an RF signal via theoutput 110, the RF cable 124A, the impedance matching circuit 122, andthe RF transmission line 128 to the ESC 132, the voltage and currentprobe 108 measures the complex voltage and current at the output 110.Similarly, when the y MHz generator generates and provides an RF signalvia the output 114, the RF cable 124B, and the RF transmission line 128to the ESC 132, the voltage and current probe 112 measures the complexvoltage and current at the output 114. Also, when the z MHz generatorgenerates and provides an RF signal via the output 118, the RF cable124C, and the RF transmission line 128 to the ESC 132, the voltage andcurrent probe 116 measures the complex voltage and current at the output118.

The complex voltages and currents measured by the voltage and currentprobes 108, 112, and 116 are provided via corresponding communicationdevices 140A, 140B, and 140C from the corresponding voltage and currentprobes 108, 112, and 116 via a processor 142 of a host system 143 to astorage hardware unit (HU) 144 of the host system 143 for storage. Forexample, the complex voltage and current measured by the voltage andcurrent probe 108 is provided via the communication device 140A and acable 142A to the processor 142, the complex voltage and currentmeasured by the voltage and current probe 112 is provided via thecommunication device 140B and a cable 142B to the processor 142, and thecomplex voltage and current measured by the voltage and current probe116 is provided via the communication device 140C and a cable 142C tothe processor 142. The processor 142 stores the complex voltages andcurrent received from the communication devices 140A, 140B, and 140C inthe storage HU 144. Examples of a communication device include anEthernet device that converts data into Ethernet packets and convertsEthernet packets into data, an Ethernet for Control AutomationTechnology (EtherCAT) device, a serial interface device that transfersdata in series, a parallel interface device that transfers data inparallel, a Universal Serial Bus (USB) interface device, etc.

Examples of the host system 143 include a computer, e.g., a desktop, alaptop, a tablet, etc. As used herein, the processor 142 may be acentral processing unit (CPU), a microprocessor, an application specificintegrated circuit (ASIC), a programmable logic device (PLD), etc.Examples of the storage HU 144 include a read-only memory (ROM), arandom access memory (RAM), or a combination thereof. The storage HU 144may be a flash memory, a redundant array of storage disks (RAID), a harddisk, etc.

The impedance matching model 102 is generated by the processor 142 andis stored within the storage HU 144. In some embodiments, the processor142 receives the impedance matching model 102 from another processor.The impedance matching model 102 represents the impedance matchingcircuit 122. For example, the impedance matching model 102 has similarcharacteristics, e.g., capacitances, inductances, resistances, complexpower, complex voltage and currents, impedance, a combination thereof,etc., as that of the impedance matching circuit 122. To illustrate, theimpedance matching model 102 has the same number of capacitors,resistors, and/or inductors as that within the impedance matchingcircuit 122, and the capacitors, resistors, and/or inductors areconnected with each other in the same manner, e.g., serial, parallel,etc. as that within the impedance matching circuit 122. In thisillustration, the impedance matching model 102 has the same capacitance,or resistance, or inductance, or a combination thereof, etc., as acapacitance, or a resistance, or an inductance, or a combinationthereof, etc., of the impedance matching circuit 122. To provide anillustration, when the impedance matching circuit 122 includes acapacitor coupled in series with an inductor, the impedance matchingmodel 102 also includes a capacitor coupled in series with an inductor.

To further illustrate, the impedance matching circuit 122 includes oneor more electrical components and the impedance matching model 102includes a design, e.g., a computer-generated model, of the impedancematching circuit 122. The computer-generated model may be generated bythe processor 142 based upon input signals received from a user via aninput HU. The input signals include signals regarding which electricalcomponents, e.g., capacitors, inductors, etc., to include in a model anda manner, e.g., series, parallel, etc., of coupling the electricalcomponents with each other. To illustrate, the impedance circuit 122includes hardware electrical components and hardware connections betweenthe electrical components and the impedance matching model 102 includessoftware representations of the hardware electrical components and ofthe hardware connections. To provide yet another illustration, theimpedance matching model 102 is designed using a software program andthe impedance matching circuit 122 is made on a printed circuit board.As used herein, electrical components may include resistors, capacitors,inductors, connections between the resistors, connections between theinductors, connections between the capacitors, and/or connectionsbetween a combination of the resistors, inductors, and capacitors.

To provide another illustration, the impedance matching model 102 isrepresented by a function as that used to represent the impedancematching circuit 122. For example, the impedance matching model 102 isrepresented by a function, e.g., a mathematical function, etc., ofresistances and reactances, and the function represents the impedancematching circuit 122.

The cable models 104A, 104B, 104C, and the RF transmission model 106 aregenerated by the processor 142 and are stored in the storage HU 144. Insome embodiments, the cable models 104A, 104B, 104C, and the RFtransmission model 106 are received by the processor 142 from anotherprocessor.

The cable model 104A represents the RF cable 124A, the cable model 104Brepresents the RF cable 124B, and the cable model 104C represents the RFcable 124C. For example, the cable model 104A and the RF cable 124A hassimilar characteristics, a cable model 104B and the RF cable 124B hassimilar characteristics, and a cable model 104C and the RF cable 124Chas similar characteristics. For example, the cable model 104B has thesame number of circuit elements, e.g., resistors, capacitors and/orinductors, etc., as that within the RF cable 124A, and the resistors,capacitors and/or inductors are connected with each other in the samemanner, e.g., serial, parallel, etc. as that within the RF cable 124A.As another example, an inductance, a capacitance, or a combinationthereof, etc., of the cable model 104A is the same as an inductance, acapacitance, or a combination thereof, etc., of the RF cable 124A. Asanother example, the cable model 104A is a computer-generated model ofRF cable 124A, the cable model 104B is a computer-generated model of theRF cable 124B, and the cable model 104C is a computer-generated model ofthe RF cable 124C. As yet another example, the cable model 104A isrepresented by a function, e.g., a mathematical function, etc., ofresistances and reactances, and the function represents the RF cable124A. As another example, the cable model 104B is represented by afunction, e.g., a mathematical function, etc., of resistances andreactances, and the function represents the RF cable 124B. As anotherexample, the cable model 104C is represented by a function, e.g., amathematical function, etc., of resistances and reactances, and thefunction represents the RF cable 124C. The cable model 104A has an input105A, the cable model 104B has an input 105B, and the cable model 104Chas an input 105C.

The RF transmission model 106 represents the RF transmission line 128.For example, the RF transmission model 106 and the RF transmission line128 have similar characteristics. As another example, the RFtransmission model 106 has the same number of circuit elements, e.g.,resistors, capacitors and/or inductors, etc., as that within the RFtransmission line 128, and the resistors, capacitors and/or inductorsare connected with each other in the same manner, e.g., serial,parallel, etc. as that within the RF transmission line 128. To furtherillustrate, when the RF transmission line 128 includes a capacitorcoupled in parallel with a resistor, the RF transmission model 106 alsoincludes the capacitor coupled in parallel with the resistor. As yetanother example, the RF transmission line 128 includes one or moreelectrical components and the RF transmission model 106 includes adesign, e.g., a computer-generated model, of the RF transmission line128. As another example, the RF transmission model 106 is represented bya function, e.g., a mathematical function, etc., of resistances andreactances, and the function represents the RF transmission line 128. Asanother example, an impedance, an inductance, a capacitance, or acombination thereof, etc., of the RF transmission model 106 is the sameas an impedance, inductance, a capacitance, or a combination thereof,etc., of the RF transmission line 128.

In some embodiments, the RF transmission model 106 is acomputer-generated impedance transformation involving computation ofcharacteristics, e.g., capacitances, resistances, inductances, acombination thereof, etc., of elements, e.g., capacitors, inductors,resistors, a combination thereof, etc., and determination ofconnections, e.g., series, parallel, etc., between the elements.

The processor 142 generates the impedance matching model 102 andconverts, e.g., segments, etc., the impedance matching model 102 intoone or more modules. Similarly, the processor 142 generates the cablemodel 104A and converts, segments, etc., the cable model 104A into oneor more modules, generates the cable model 104B and converts the cablemodel 104B into one or more modules, and generates the cable model 104Cand segments the cable model 104C into one or more modules. Moreover,the processor 142 generates the RF transmission model 106 and converts,segments, etc., the RF transmission model 106 into one or more modules.

Based on the complex voltage and current received at the input 105A fromthe voltage and current probe 108 via the cable 142A andcharacteristics, e.g., impedance, resistance, reactance, complex voltageand current, etc., of the one or more modules of the cable model 104A,the processor 142 calculates a complex voltage and current at an input146A of the impedance matching model 102. The complex voltage andcurrent at the input 146A is stored in the storage HU 144.

Similarly, based on the complex voltage and current received at theinput 105B from the voltage and current probe 112 via the cable 142B andcharacteristics, e.g., impedance, resistance, reactance, complex voltageand current, etc., of the one or more modules of the cable model 104B,the processor 142 calculates a complex voltage and current at an input146B of the impedance matching model 102. Also, based on the complexvoltage and current received at the input 105C from the voltage andcurrent probe 116 via the cable 142C and characteristics, e.g.,impedance, resistance, reactance, complex voltage and current, etc., ofthe one or more modules of the cable model 104C, the processor 142calculates a complex voltage and current at an input 146C of theimpedance matching model 102.

Moreover, based on the complex voltage and current at the input 146A andcharacteristics, e.g., impedance, resistance, reactance, complex voltageand current, etc., of the one or more modules of the impedance matchingmodel 102, the processor 142 calculates a complex voltage and current atan output 148 of the impedance matching model 102. A complex voltage andcurrent at the output 148 is stored in the storage HU 144.

Similarly, based on the complex voltage and current at the input 146Band characteristics, e.g., impedance, resistance, reactance, complexvoltage and current, etc., of the one or more modules of the impedancematching model 102, the processor 142 calculates a complex voltage andcurrent at the output 148 of the impedance matching model 102. Also,based on the complex voltage and current at the input 146C andcharacteristics, e.g., impedance, resistance, reactance, complex voltageand current, etc., of the one or more modules of the impedance matchingmodel 102, the processor 142 calculates a complex voltage and current atthe output 148 of the impedance matching model 102.

In some embodiments, a voltage magnitude is a root mean square (RMS)voltage and a current magnitude is an RMS current.

The output 148 is coupled to an input of the RF transmission model 106,which is stored in the storage HU 144.

Based on the complex voltage and current at the output 148 andcharacteristics, e.g., impedance, resistance, reactance, complex voltageand current, etc., of the one or more modules of the RF transmissionmodel 106, the processor 142 calculates a complex voltage and current atan output 150 of the RF transmission model 106. The output 150 is amodel of an output 151 of the RF transmission line 128 and the output151 is coupled to the ESC 132 to provide RF signals generated by one ormore of the x, y, and z MHz RF generators to the ESC 132. The complexvoltage and current determined at the output 150 is stored in thestorage HU 144.

It should be noted that although three generators are shown coupled tothe impedance matching circuit 122, in one embodiment, any number of RFgenerators, e.g., a single generator, two generators, etc., are coupledto the plasma chamber 130 via an impedance matching circuit.

It should further be noted that although the above embodiments aredescribed with respect to using a complex voltage and current, insteadof the complex voltage and current, the embodiments may be describedusing impedances. For example, based on an impedance determined from thecomplex voltage and current received from the voltage and current probe108 via the cable 142A and the one or more modules of the cable model104A, the processor 142 calculates an impedance at the input 146A of theimpedance matching model 102. The impedance is determined by theprocessor 142 from the complex voltage and current received from thevoltage and current probe 108. As another example, based on theimpedance at the input 146A and the one or more modules of the impedancematching model 102, the processor 142 calculates an impedance at theoutput 148 of the impedance matching model 102. As yet another example,based on the impedance at the output 148 and the one or more modules ofthe RF transmission model 106, the processor 142 calculates an impedanceat the output 150 of the RF transmission model 106.

FIG. 2 is a diagram of an embodiment of an impedance matching model 103after conversion, e.g., segmentation, etc. of the impedance matchingmodel 102 (FIG. 1). The processor 142 (FIG. 1) segments the impedancematching model 102 into multiple modules 201, 203, and 205. In someembodiments, the processor 142 segments the impedance matching model 102into any number of modules, e.g., N modules, where N is an integergreater than zero.

The processor 142 maintains a coupling between elements of the impedancematching model 102 after the segmentation of the impedance matchingmodel 102 into modules 201, 203, and 205. For example, the processormaintains a series connection or a parallel connection between twocircuit elements, e.g., a capacitor and an inductor, a resistor and aninductor, a capacitor and a resistor, etc., of the impedance matchingmodel 102 before and after the segmentation.

The modules 201, 203, and 205 of the impedance matching model 103 arecoupled with each other. For example, the module 201 is coupled to themodule 203 via a link 202 and the module 203 is coupled to the module205 via a link 204.

The module 201 has an input 206, which is an example of the input 146A,the input 146B, or the input 146C (FIG. 1) of the impedance matchingmodel 102. The module 201 has an output 208, which is coupled to aninput 210 of the module 203. The module 203 has an output 212, which iscoupled to an input 214 of the module 205. The module 205 has an output216, which is an example of the output 148 (FIG. 1) of the impedancematching model 102.

To generate an impedance matching model of another impedance matchingcircuit (not shown), e.g., a circuit that is other than and thatreplaces the impedance matching circuit 122 (FIG. 1), the processor 142replaces the module 201 with another module (not shown), replaces themodule 203 with another module (not shown), and/or replaces the module205 with another module (not shown). The processor 142 establishes aseries link between the replacement modules and unreplaced modules,e.g., the module 201, 203, or 205, etc., when all of the modules 201,203 and 205 are not replaced or establishes a series link between thereplacement modules when all of the modules 201, 203 and 205 arereplaced by the replacement modules.

A series combination of the replacement modules (not shown) thatreplaces the corresponding modules 201, 203, and/or 205 has similarcharacteristics as that of the other impedance matching circuit (notshown). For example, a combined impedance of the replacement modules(not shown) is the same as or within a range of an impedance of theother impedance matching circuit (not shown). In this example, thereplacement modules (not shown) represent the other impedance matchingcircuit (not shown). As another example, a combined impedance of one ofthe replacement modules (not shown), the module 203, and the module 205is the same as or within a range of an impedance of the other impedancematching circuit (not shown). In this example, the one of thereplacement modules (not shown), the module 203, and the module 205represent the other impedance matching circuit (not shown). Modularityof impedance matching models allows easy replacement of one or moremodules of one of the impedance matching models with one or more modulesof another one of the impedance matching models.

Upon replacing the module 201 with another module (not shown), replacingthe module 203 with another module (not shown), and/or replacing themodule 205 with another module, the processor 142 checks whethercharacteristics, e.g., impedance, complex voltage and current, etc., ofan impedance matching model that includes one or more of the replacementmodules (not shown) and/or one or more of the modules 201, 203, and 205are similar to characteristics, e.g., impedance, complex voltage andcurrent, etc., of the other impedance matching circuit (not shown). Forexample, the processor 142 calculates a combined impedance of thereplacement modules (not shown) and/or one or more of the modules 201,203, and 205 and compares the combined impedance with an impedance ofthe other replacement impedance matching circuit (not shown). Upondetermining that the combined impedance of the replacement modules (notshown) and/or one or more of the modules 201, 203, and 205 matches withor is within a range of the impedance of the other replacement impedancematching circuit (not shown), the processor 142 determines thatcharacteristics of the impedance matching model that includes one ormore of the replacement modules (not shown) and/or one or more of themodules 201, 203, and 205 are similar to characteristics of the otherimpedance matching circuit (not shown). On the other hand, upondetermining that the combined impedance of the replacement modules (notshown) and/or one or more of the modules 201, 203, and 205 do not matchwith or is not within a range of the impedance of the other replacementimpedance matching circuit (not shown), the processor 142 determinesthat characteristics of the impedance matching model that includes oneor more of the replacement modules (not shown) and/or one or more of themodules 201, 203, and 205 are not similar to characteristics of theother impedance matching circuit (not shown).

In various embodiments, the impedance of the other replacement impedancematching circuit is received by the processor 142 from anotherprocessor. In some embodiments, the impedance of the other replacementimpedance matching circuit is calculated by the processor 142 based oncomplex voltages and currents measured at an input and at an output ofthe other replacement impedance matching circuit.

FIG. 3 is a diagram of an embodiment of a module n of the impedancematching model 103, where n ranges from 1 thru N. The module n includesa series circuit 218 and a shunt circuit 220. In some embodiments, themodule n includes only one series circuit 218 and only one shunt circuit220. The shunt circuit 220 is coupled to a ground connection 222. Also,the parallel shunt circuit 220 is coupled to the series circuit 218.

The module n has an input 224, which is an example of the input 206, theinput 210, or the input 214 (FIG. 2). Moreover, the module n has anoutput 226, which is an example of the output 208, the output 212, orthe output 216 (FIG. 2).

As shown, the series circuit 218 is coupled to the input 224 and to theoutput 226. Moreover, the shunt circuit 220 is coupled to the output226.

In some embodiments, a quadratic function is used instead of the seriescircuit 218 and a quadratic function is used instead of the shuntcircuit 220. The quadratic function that is used instead of the seriescircuit 218 represents a directional sum of resistances of all elementsof the series circuit 218 and a directional sum of reactances of theelements of the series circuit. For example, the series circuit isrepresented as R_(s)+jX_(s), where R_(s) is a result of a directionalsum of resistances of all elements of the series circuit 218, X_(s) is aresult of a directional sum of reactances of all elements of the seriescircuit 218, and j is the imaginary unit. Moreover, the quadraticfunction that is used instead of the shunt circuit 220 represents adirectional sum of resistances of all elements of the shunt circuit 220and a directional sum of reactances of all elements of the shunt circuit220. For example, the shunt circuit is represented as R_(p)+jX_(p),where R_(p) is a result of a directional sum of resistances of allelements of the shunt circuit 220, and X_(p) is a result of adirectional sum of reactances of all elements of the shunt circuit 220.

In various embodiments, the series circuit 218 or the shunt circuit 220includes a resistor coupled in series with an inductor and a capacitor.In some embodiments, the series circuit 218 or the shunt circuit 220includes a resistor coupled in series with an inductor or in series witha capacitor. In several embodiments, the series circuit 218 or the shuntcircuit 220 includes an inductor coupled in series with a capacitor. Inseveral embodiments, the series circuit 218 or the shunt circuit 220includes an inductor, a resistor, or a capacitor.

In some embodiments, the processor 142 (FIG. 1) determines an impedanceZ_((n+1)-in) at an input of an (n+1)^(th) module of the impedancematching model 103 (FIG. 2) based on an impedance Z_(n-in) at an inputof the n^(th) module of the impedance matching model 103 andcharacteristics, e.g., parameters, etc., of the n^(th) module. Forexample, the processor 142 determines the impedance Z_((n+1)-in)according to a function:

$\begin{matrix}{Z_{{({n + 1})} - {i\; n}} = \frac{Z_{np}( {Z_{n - {i\; n}} - Z_{n\; s}} )}{Z_{np} - ( {Z_{n - {i\; n}} - Z_{n\; s}} )}} & (1)\end{matrix}$

where Z_(np) is an impedance of the shunt circuit 220 and Z_(ns) is animpedance of the series circuit 218, and where Z_(np) and Z_(ns) areparameters of the n^(th) module. The (n+1)^(th) module follows and isconsecutive to the n^(th) module. For example, when the module 201 (FIG.2) is the n^(th) module, the module 203 (FIG. 2) is the (n+1)^(th)module.

In several embodiments, when the n^(th) module is a first module of theimpedance matching model 103, the processor 142 determines the impedanceZ_(n-in) at an input of the n^(th) module based on an impedance at theoutput 110 (FIG. 1) of the x MHz RF generator and characteristics of thecable model 104A (FIG. 1). For example, the processor 142 calculates animpedance of the cable model 104A based on elements of the cable model104A and generates a directional sum of impedance generated from complexvoltage and current measured at the output 110 and the impedance of thecable model 104A.

In some embodiments, an impedance at an output of an RF generator is aload impedance as seen by the generator. For example, an impedance atthe output 110 of the x MHz RF generator is a load impedance as seen bythe x MHz RF generator.

In various embodiments, the processor 142 (FIG. 1) determines a powerP_(loss-n), which is power lost in the n^(th) module based on a powerP_(n-in), which is power input to the n^(th) module and parameters ofthe n^(th) module. For example, the processor 142 determines the powerloss P_(loss-n) according to a function:

$\begin{matrix}{P_{{loss} - n} = {P_{n - {i\; n}}\lbrack {\frac{{Re}( Z_{n\; s} )}{{Re}( Z_{n - {i\; n}} )} + \{ {\frac{{Re}( Z_{np} )}{{Re}( Z_{n - {i\; n}} )}{\frac{Z_{n - {i\; n}} - Z_{n\; s}}{Z_{np}}}^{2}} \}} \rbrack}} & (2)\end{matrix}$

where Re(Z_(ns)) is a resistance of the impedance Z_(ns), Re(Z_(n-in))is a resistance of the impedance Z_(n-in), and Re(Z_(np)) is aresistance of the impedance Z_(np), and “| |” represents a magnitude ofimpedance. In various embodiments, the processor 142 subtracts the powerloss P_(loss-n) from the input power P_(n-in) to determine powerP_((n+1)-in) that is input to the consecutive (n+1)^(th) module.

In some embodiments, the power P_(n-in) input to the n^(th) module isdetermined based on the complex voltage and current measured at theoutput 110 (FIG. 1) and impedance of the cable model 104A, 104B, or 104C(FIG. 1) that is coupled to the n^(th) module.

When there are N modules in the impedance matching model 103, theprocessor 142 determines a current I_(n-out), e.g., root mean squarecurrent, current magnitude, etc., at an output of the n^(th) modulebased on the power P_(n-in) and the impedance Z_(n-in) of the n^(th)module. For example, the processor 142 determines the current I_(n-out)as a square root of a ratio of the power P_(n-in) and a resistance ofthe impedance Z_(n-in). Moreover, when there are N modules in theimpedance matching model 103, the processor 142 determines a voltageV_(n-out), e.g., root mean square voltage, voltage magnitude, etc., atan output of the n^(th) module based on the current I_(n-out) and theimpedance Z_(n-in). For example, the processor 142 calculates thevoltage V_(n-out) as a product of the current I_(n-out) and a magnitudeof the impedance Z_(n-in).

FIG. 4A is a diagram of an embodiment of a module 230, which is anexample of the module n (FIG. 3). The module 230 includes a seriesresistor-inductor-capacitor (RLC) circuit 232 and a shunt RLC circuit234. The series RLC circuit 232 is an example of the series circuit 218and the shunt RLC circuit 234 is an example of the shunt circuit 220.

The series RLC circuit 232 includes a resistor R_(fs), an inductorL_(fs), and a capacitor C_(fs). The resistor R_(fs) is coupled in serieswith the inductor L_(fs), and the inductor L_(fs) is coupled in serieswith the capacitor C_(fs). The parallel RLC circuit 234 includes aresistor R_(fp), an inductor L_(fp), and a capacitor C_(fp). Theresistor R_(fp) is coupled in series with the inductor L_(fp), and theinductor L_(fp) is coupled in series with the capacitor C_(fp). Thecapacitor C_(fp) is coupled to a ground connection 236.

Inductances of the inductor L_(fs) and L_(fp) are fixed, e.g., constant.Similarly, capacitances of the capacitors C_(fs) and C_(fp) are fixed.Also, resistances of the resistors R_(fs) and R_(fp) are fixed.

FIG. 4B is a diagram of an embodiment of a module 240 in whichinductances of inductors L_(vs) and L_(vp) are variable, e.g. not fixed.The module 240 is an example of the module n (FIG. 3). The module 240includes a series resistor-inductor-capacitor (RLC) circuit 242 and aparallel RLC circuit 244. The series RLC circuit 242 is an example ofthe series circuit 218 and the parallel RLC circuit 244 is an example ofthe shunt circuit 220 (FIG. 3). The series RLC circuit 242 includes theresistor R_(fs), the variable inductor L_(vs), and the capacitor C_(fs).The parallel RLC circuit 244 includes the resistor R_(fp), the variableinductor L_(vp), and the capacitor C_(fp). The module 240 is the same asthe module 230 (FIG. 4A) except that in the module 240, the fixedinductor L_(fs) is replaced with the variable inductor L_(vs) and thefixed inductor L_(fp) is replaced with the variable inductor L_(vp).

FIG. 4C is a diagram of an embodiment of a module 250 in whichcapacitances of capacitors C_(vs) and C_(vp) are variable. The module250 is an example of the module n (FIG. 3). The module 250 includes aseries resistor-inductor-capacitor (RLC) circuit 252 and a parallel RLCcircuit 254. The series RLC circuit 252 is an example of the seriescircuit 218 and the parallel RLC circuit 254 is an example of the shuntcircuit 220 (FIG. 3). The series RLC circuit 252 includes the resistorR_(fs), the fixed inductor L_(fs), and the variable capacitor C_(vs).The parallel RLC circuit 254 includes the resistor R_(fp), the fixedinductor L_(fp), and the variable capacitor C_(vp). The module 250 isthe same as the module 230 (FIG. 4A) except that in the module 250, thefixed capacitor C_(fs) is replaced with the variable capacitor C_(vs)and the fixed capacitor C_(fp) is replaced with the variable capacitorC_(vp).

FIG. 4D is a diagram of an embodiment of a module 260 in whichcapacitances of capacitors C_(vs) and C_(vp) are variable andinductances of the inductors L_(vs) and L_(vp) are variable. The module260 is an example of the module n (FIG. 3). The module 260 includes aseries resistor-inductor-capacitor (RLC) circuit 262 and a parallel RLCcircuit 264. For example, the series RLC circuit 262 is an example ofthe series circuit 218 and the parallel RLC circuit 264 is an example ofthe shunt circuit 220. The series RLC circuit 262 includes the resistorR_(fs), the variable inductor L_(vs), and the variable capacitor C_(vs).The parallel RLC circuit 264 includes the resistor R_(fp), the variableinductor L_(vp), and the variable capacitor C_(vp). The module 260 isthe same as the module 230 (FIG. 4A) except that in the module 260, thefixed capacitor C_(fs) is replaced with the variable capacitor C_(vs),the fixed capacitor C_(fp) is replaced with the variable capacitorC_(vp), the fixed inductor L_(fs) is replaced with the variable inductorL_(vs), and the fixed inductor L_(fp) is replaced with the variableinductor L_(vp).

In some embodiments, a value of resistance of the resistor R_(fs) iszero and/or a value of resistance of the resistor R_(fp) is zero. Invarious embodiments, a value of inductance of the inductor L_(fs) iszero, a value of inductance of the inductor L_(vs) is zero, a value ofinductance of the inductor L_(fp) is zero, and/or a value of inductanceof the inductor L_(vp) is zero. In some embodiments, a value ofcapacitance of the capacitor C_(fs) is zero, a value of capacitance ofthe capacitor C_(vs) is zero, a value of capacitance of the capacitorC_(fp) is zero, and/or a value of capacitance of the capacitor C_(vp) iszero.

FIG. 4E is a diagram of an embodiment of a module 270 that represents afunction 272 that is implemented within the series circuit 218 (FIG. 3)and a function 274 that is implemented within the shunt circuit 220(FIG. 3). The function 272 is a mathematical function R_(s)+jX_(s) andthe function 274 is a mathematical function R_(p)+jX_(p). The function274 is a shunt function that shunts a current output by the function272.

The processor 142 (FIG. 1) calculates the resistance R_(s) based on acenter frequency, e.g., theoretical frequency, etc., of the x MHz RFgenerator, based on an actual, e.g., measured, etc., frequency of the xMHz RF generator, and based on one or more coefficients. For example,the processor 142 calculates the resistance R_(s) as a function:R _(S) =A _(S0) +A _(S1)(F−F ₀)+A _(S2)(F−F ₀)²   (3)

where A_(s0), A_(s1), and A_(s2) are coefficients, F₀ is a centerfrequency of the x MHz RF generator and F is an actual frequency of thex MHz RF generator. In some embodiments, the processor 142 determinesthe center frequency F₀ as a frequency of a complex voltage and currentmeasured at the output 110 (FIG. 1). The processor 142 uses thecoefficients A_(s0), A_(s1), and A_(s2) that may be determined byexperimentation. For example, another processor (not shown) maydetermine the coefficients A_(s0), A_(s1), and A_(s2) by, e.g.,determining the actual frequency of the x MHz RF generator for a numberof times complex voltages and currents measured at the output 110(FIG. 1) are received from the voltage and current probe 108,determining resistances at a point within the impedance matching circuit122 (FIG. 1) corresponding to an output of the function 272 of then^(th) module for the number of times, and solving the function (3) fora fit, e.g., best fit, linear fit, etc., for the coefficients A_(s0),A_(s1), and A_(s2). The processor 142 receives the coefficients A_(s0),A_(s1), and A_(s2) from the other processor (not shown).

The processor 142 calculates the reactance X_(s) based on the centerfrequency of the x MHz RF generator, based on the actual frequency ofthe x MHz RF generator, and based on one or more coefficients. Forexample, the processor 142 calculates the reactance X_(s) as a function:X _(S) =B _(S0) +B _(S1)(F−F ₀)+B _(s2)(F−F ₀)²   (4)

where B_(s0), B_(s1), and B_(s2) are coefficients. The processor 142uses the coefficients B_(s0), B_(s1), and B_(s2) that may be determinedby experimentation. For example, another processor (not shown) maydetermine the coefficients B_(s0), B_(s1), and B_(s2) by e.g.,determining the actual frequency of the x MHz RF generator for a numberof times complex voltages and currents measured at the output 110(FIG. 1) are received from the voltage and current probe 108,determining reactances at a point within the impedance matching circuit122 (FIG. 1) corresponding to an output of the function 272 of then^(th) module for the number of times, and solving the function (4) fora fit, e.g., best fit, linear fit, etc., for the coefficients B_(s0),B_(s1), and B_(s2). The processor 142 receives the coefficients B_(s0),B_(s1), and B_(s2) from the other processor (not shown).

The processor 142 calculates the resistance R_(p) based on the centerfrequency of the x MHz RF generator, based on the actual frequency ofthe x MHz RF generator, and based on one or more coefficients. Forexample, the processor 142 calculates the resistance R_(p) as afunction:R _(P) =A _(p0) +A _(p1)(F−F ₀)+A _(p2)(F−F ₀)²   (5)

where A_(p0), A_(p1), and A_(p2) are coefficients. The processor 142receives the coefficients A_(p0), A_(p1), and A_(p2) from anotherprocessor (not shown) that determines the coefficients A_(p0), A_(p1),and A_(p2) in a similar manner as that described above.

The processor 142 calculates the reactance X_(p) based on the centerfrequency of the x MHz RF generator, based on the actual frequency ofthe x MHz RF generator, and based on one or more coefficients. Forexample, the processor 142 calculates the reactance X_(p) as a function:X _(p) =B _(p0) +B _(p1)(F−F ₀)+B _(p2)(F−F ₀)²   (6)

where B_(p0), B_(p1), and B_(p2) are coefficients. The processor 142receives the coefficients B_(p0), B_(p1), and B_(p2) from anotherprocessor (not shown) that determines the coefficients B_(p0), B_(p1),and B_(p2) in a similar manner as that described above.

In some embodiments, the processor 142 determines impedances, e.g.,resistances, reactances, etc., at a point within the impedance matchingcircuit 122 (FIG. 1) corresponding to the output of the function 272 orto the output of the function 274 of the n^(th) module based on acomplex voltage and current measured by a voltage and current probe (notshown) at the point. In various embodiments, the point within theimpedance matching circuit 122 (FIG. 1) corresponds to the output of thefunction 272 or to the output of the function 274 of the n^(th) modulewhen an impedance between an input of the impedance matching circuit 122and the point within the impedance matching circuit 122 is the same asor within a range of an impedance between an input of the impedancematching module 102 and the point within the impedance matching module102 and when an impedance between an output of the impedance matchingcircuit 122 and the point within the impedance matching circuit 122 isthe same as or within a range of an impedance between an output of theimpedance matching module 102 and the point within the impedancematching module 102.

In some embodiments, an impedance of a circuit element of a model isequal to a resistance of the circuit element when a reactance of thecircuit element is zero. In various embodiments, an impedance of acircuit element of a model is equal to a reactance of the circuitelement when a resistance of the circuit element is zero.

In the embodiments described with reference to FIG. 4E, the groundconnection 222 (FIG. 3) is referred to as a ground function.

FIG. 5A is a diagram of a circuit 300, which is an example of theimpedance matching model 102 (FIG. 2). The circuit 300 is divided by theprocessor 142 (FIG. 1) into an x MHz matching model 302, a y MHzmatching model 306, and a z MHz matching model 308. The x MHz matchingmodel 302 includes elements, e.g., a capacitor C1, a capacitor C2, aninductor L1, a capacitor C3, and an inductor L2, etc., coupled to aninput 304A, which is an example of the input 146A (FIG. 1). The y MHzmatching model 306, which includes elements, e.g., an inductor L3, acapacitor C4, a capacitor C5, and an inductor L4, etc., coupled to aninput 304B, which is an example of the input 146B (FIG. 1). Moreover,the z MHz matching model 308, which includes elements, e.g., an inductorL5, a capacitor C6, a capacitor C7, and an inductor L6, an inductor L7,a capacitor C8, etc., coupled to an input 304C, which is an example ofthe input 146C (FIG. 1).

In some embodiments, the x MHz model 302 receives a complex voltage andcurrent from the probe 108 (FIG. 1) of an RF signal that has afrequency, e.g., an operating frequency of the x MHz RF generator, etc.,ranging between 1.8 and 2.17 MHz of the x MHz RF generator. In variousembodiments, the y MHz model 306 receives a complex voltage and currentfrom the probe 112 (FIG. 1) of an RF signal that has a frequency, e.g.,an operating frequency of the y MHz RF generator, etc., ranging between25.7 and 28.5 MHz of the y MHz RF generator. In some embodiments, the zMHz model 308 receives a complex voltage and current from the probe 116(FIG. 1) of an RF signal that has a frequency, e.g., an operatingfrequency of the z MHz RF generator, etc., ranging between 57 and 60 MHzof the z MHz RF generator.

In several embodiments, the x MHz matching model 302 includes any numberof inductors, any number of capacitors, and/or any number of resistors.In some embodiments, the y MHz matching model 306 includes any number ofinductors, any number of capacitors, and/or any number of resistors. Inseveral embodiments, the z MHz matching model 308 includes any number ofinductors, any number of capacitors, and/or any number of resistors. Forexample, the circuit 300 may be changed to include resistive losses inone or more of the capacitors C1, C2, C3, C4, C5, C6, C7, and C8. Asanother example, the circuit 300 may be changed to include resistivelosses in one or more of the inductors L1, L2, L3, L4, L5, L6, and L7.As yet another example, the circuit 300 may be changed to includevariable inductance of one or more of the capacitors C1, C2, C3, C4, C5,C6, C7, and C8. As another example, the circuit 300 may be changed toinclude variable capacitance of one or more of the inductors L1, L2, L3,L4, L5, L6, and L7. As another example, the circuit 300 may be changedto include a stray capacitance to a ground connection. As yet anotherexample, the circuit 300 may be changed to include a capacitance and/oran inductance of an RF strap of the RF transmission line 128. As anotherexample, the circuit 300 may be changed to consider finite length of oneor more of the inductors L1, L2, L3, L4, L5, L6, and L7 and the finitelength is not negligible compared to a wavelength of an RF signal thattransfers through the inductor.

FIG. 5B is a diagram of an embodiment of a segmented circuit 400, whichis an example of the impedance matching model 103 (FIG. 2). Theprocessor 142 segments the circuit 300 into modules 402, 404, 406, and408 to generate the segmented circuit 400. For example, the processor142 segments the z MHz impedance model 308 (FIG. 5A) into the module402, the module 404, the module 406, and allocates the inductor L7 tothe module 408. Moreover, the processor 142 combines the x MHz impedancemodel 302, the y MHz impedance model 306, and the inductor L7 into themodule 408.

The module 402 includes the inductor L5, which is a shunt circuit thatacts a shunt to a series circuit 410. Moreover, the module 404 includesthe capacitor C6. The series circuit 410 and the inductor L5 are coupledto an output 414 of the module 402. The series circuit 412 and thecapacitor C6 are coupled to an output 415 of the module 404. The seriescircuit 412 is also coupled to the output 414 of the module 402.

The module 406 includes a series circuit 416 that includes the capacitorC7 and the inductor L6. The inductor L6 is coupled in series to thecapacitor C7. Moreover, the module 406 includes the capacitor C8. Theseries circuit 416 and the capacitor C8 are coupled to an output 417 ofthe module 406. The series circuit 416 is also coupled to the output 415of the module 404.

Also, the module 408 includes a series circuit 418 that includes theinductor L7. The module 408 includes a shunt circuit 420 that includesthe inductors L1, L2, L3, L4, and the capacitors C1, C2, C3, C4, and C5.The circuit 420 acts as shunt to the series circuit 418. The seriescircuit 418 is coupled to the output 417 of the module 406. Also, theseries circuit 418 and the shunt circuit 420 are coupled to an output419, which is an example of the output 216 (FIG. 2). The shunt circuit420 is coupled to the inputs 304A and 304B.

FIG. 5C is a diagram of an embodiment of a segmented circuit 500 that isgenerated from the segmented circuit 400 (FIG. 5B). The segmentedcircuit 500 is an example of the impedance matching model 103 (FIG. 2).The segmented circuit 500 includes the modules 402, 404, and 406, andincludes a module 502. The module 502 includes the series circuit 418and a shunt circuit 506. The shunt circuit 506 includes a resistor R_(C)and an inductor L_(c) in series with a capacitor C_(c). The processor142 determines a combined impedance of the L1, L2, L3 and L4 and thecapacitors C1, C2, C3, C4, and C5 and the inputs 304A and 304B and thecombined impedance is represented by the processor 142 as a combinationof the resistor R_(C), the inductor Lc, and the capacitor C_(C).

In some embodiments, a combined capacitance of two capacitors inparallel with each other having positively charged plates coupled to aninput wire and negatively charged plates coupled to an output wire is asum of capacitances of the two capacitors. In various embodiments, acombined capacitance of two capacitors in series with each other havinga positively charged plate of a first one of the two capacitors coupledto a negatively charged plate of a second one of the two capacitors isequal to a ratio of a product of capacitances of the two capacitors to asum of the two capacitances.

In various embodiments, a combined inductance of two inductors in serieswith each other having a positively charged terminal of a first one ofthe two inductors coupled to a negatively charged terminal of a secondone of the two inductors is equal to a sum of inductances of the twoinductors. In various embodiments, a combined inductance of twoinductors in parallel with each other having a positively chargedterminal of a first one of the two inductors coupled to a negativelycharged terminal of a second one of the two inductors is equal to aratio of product of inductances of the two inductors to a sum of theinductances of the two inductors.

In several embodiments, a combined resistance of two resistors in serieswith each other having a positively charged terminal of a first one ofthe two resistors coupled to a negatively charged terminal of a secondone of the two resistors is equal to a sum of resistances of the tworesistors. In various embodiments, a combined resistance of tworesistors in parallel with each other having a positively chargedterminal coupled to a first end of the two resistors and having anegatively charged terminal coupled to a second end of the two resistorsis equal to a ratio of product of resistances of the two resistors to asum of resistances of the two resistors.

In various embodiments, a combined impedance of an inductor and acapacitor in series with each other is a sum of an impedance of theinductor and an impedance of the capacitor. In some embodiments, acombined impedance of a resistor and a capacitor in series with eachother is a sum of an impedance of the resistor and an impedance of thecapacitor. In a number of embodiments, a combined impedance of aresistor and an inductor in series with each other is a sum of animpedance of the resistor and an impedance of the inductor.

In some embodiments, a combined impedance of an inductor and a capacitorin parallel with each other is a ratio of a product of an impedance ofthe inductor and an impedance of the capacitor over a sum of theimpedance of the inductor and the impedance of the capacitor. In variousembodiments, a combined impedance of an inductor and a resistor inparallel with each other is a ratio of a product of an impedance of theinductor and an impedance of the resistor over a sum of the impedance ofthe inductor and the impedance of the resistor. In several embodiments,a combined impedance of an inductor and a capacitor in parallel witheach other is a ratio of a product of an impedance of the inductor andan impedance of the capacitor over a sum of the impedance of theinductor and the impedance of the capacitor.

In various embodiments, the module 502 represents an effect of the x andy MHz matching models 302 and 306 (FIG. 5A) in a simplified form on thez MHz matching model 308. For example, the processor 142 (FIG. 1)generates and couples the module 502 in series with the modules 402,404, and 406 to account for an effect of impedances of the matchingmodels 302 and 306 on an impedance of the z MHz matching model 308. Asanother example, the processor 142 calculates a combined impedance ofthe modules 402, 404, 406, and 502 to account for and simplify an effectof impedances of the matching models 302 and 306 on an impedance of thez MHz matching model 308.

FIG. 6 is a diagram of an embodiment of a module 229 that is similar tothe module n of FIG. 3 except that positions of the series circuit 218and the shunt circuit 220 are changed compared to positions of theseries circuit 218 and the shunt circuit 220 in the module n. In themodule 229, the shunt circuit 220 is placed on an opposite side of theseries circuit 218 of the module 229 compared to a side on which theshunt circuit 220 is placed in the module n. The shunt circuit 220 iscoupled to the input 224 and to the series circuit 218 and the seriescircuit 218 is coupled to the output 226. Also, the series circuit 218is coupled to the input 224. Moreover, the shunt circuit 220 of themodule 229 shunts a signal that is received as an input by the seriescircuit 218 of the module 229. Comparatively, the shunt circuit 220 ofthe module n shunts a signal that is provided as an output by the seriescircuit 218 of the module n.

The module 229 is an example of any of the modules N of the impedancematching model 103 (FIG. 2). For example, the module 229 is an exampleof the module 201 or the module 203 or the module 205.

In some embodiments, the module 229 includes only one series circuit 218and only one shunt circuit 220.

FIG. 7A is a diagram of an embodiment of a module 231, which is anexample of the module 229 (FIG. 6). The series RLC circuit 232 of themodule 231 is placed on a side of the parallel circuit 234 of the module231 and the side is opposite to a side on which the series RLC circuit232 of the module 230 (FIG. 4A) is placed.

FIG. 7B is a diagram of an embodiment of a module 241, which is anexample of the module 229 (FIG. 6). As shown, the series RLC circuit 242of the module 241 is placed on a side of the parallel circuit 244 of themodule 241 and the side is opposite to a side on which the series RLCcircuit 242 of the module 230 (FIG. 4B) is placed.

FIG. 7C is a diagram of an embodiment of a module 251, which is anexample of the module 229 (FIG. 6). The series RLC circuit 252 of themodule 251 is placed on a side of the parallel circuit 254 of the module251 and the side is opposite to a side on which the series RLC circuit252 of the module 250 (FIG. 4C) is placed.

FIG. 7D is a diagram of an embodiment of a module 261, which is anexample of the module 229 (FIG. 6). As visible in FIG. 7D, the seriesRLC circuit 262 of the module 261 is placed on a side of the parallelcircuit 264 of the module 261 and the side is opposite to a side onwhich the series RLC circuit 262 of the module 260 (FIG. 4D) is placed.

FIG. 7E is a diagram of an embodiment of a module 271, which is anexample of the module 229 (FIG. 6). As shown, the function 274 ispositioned in the module 271 at a side of the function 272 of the module271 and the side is opposite to a side on which the function 274 ispositioned in the module 270 (FIG. 4E).

In the embodiments described with reference to FIG. 7E, the groundconnection 222 (FIG. 3) is referred to as a ground function.

FIG. 8 is a diagram of an embodiment of a segmented cable model or asegmented RF transmission model 600, referred to herein as a cablemodel/RF transmission model 600. The cable model/RF transmission model600 is an example of a cable model generated by converting, e.g.,segmenting, etc., the cable model 104A (FIG. 1), or a cable modelgenerated by converting the cable model 104B (FIG. 1), or a cable modelgenerated by converting the cable model 104C (FIG. 1), or an RFtransmission model generated by converting the RF transmission model 106(FIG. 1).

It should be noted that the cable model generated by converting thecable model 104A may have different number of modules than that of thecable model generated by converting the cable model 104B and that of thecable model generated by converting the cable model 104C. Similarly, thecable model generated by converting the cable model 104B may havedifferent number of modules than that of the cable model cable modelgenerated by converting the cable model 104C. Moreover, it should benoted that the RF transmission model 106 may have a different number ofmodules than that of the cable model generated by converting the cablemodel 104A, or the cable model generated by converting the cable model104B, or the cable model generated by converting the cable model 104C.The cable model/RF transmission model 600 includes one or more modules,e.g., a module 602, a module 604, and a module 606.

In some embodiments, the RF transmission model 600 is generated byconverting, e.g., segmenting, etc., the RF transmission model 106, whichis a circuit that includes one or more resistors, or one or morecapacitors, or one or more inductors, or a combination thereof. In thecircuit that includes one or more resistors, or one or more capacitors,or one or more inductors, or a combination thereof, in some embodiments,a capacitor is coupled in series or in parallel to another capacitor, aresistor, or an inductor. In the circuit that includes one or moreresistors, or one or more capacitors, or one or more inductors, or acombination thereof, in various embodiments, a resistor is coupled inseries or in parallel to another resistor, a capacitor, or an inductor.In the circuit that includes one or more resistors, or one or morecapacitors, or one or more inductors, or a combination thereof, inseveral embodiments, an inductor is coupled in series or in parallel toanother inductor, a capacitor, or a resistor.

Similarly, in various embodiments, the cable model 600 is generated byconverting, e.g., segmenting, etc., the cable model 104A, 104B, or 104C(FIG. 1), which is a circuit that includes one or more resistors, or oneor more capacitors, or one or more inductors, or a combination thereof.

The processor 142 (FIG. 1) segments the RF transmission model 106 intomultiple modules 602, 604, and 606. In some embodiments, the processor142 segments the RF transmission model 106 into any number of modules,e.g., D modules, where D is an integer greater than zero.

Similarly, in various embodiments, the processor 142 (FIG. 1) segmentsthe cable model 104A, 104B, or 104C into multiple modules 602, 604, and606. In some embodiments, the processor 142 segments the cable model104A, 104B, or 104C into any number of modules, e.g., E modules, where Eis an integer greater than zero.

The processor 142 maintains a coupling between elements of a cablemodel/RF transmission model after the segmentation of the cable model/RFtransmission model into modules 602, 604, and 606. For example, theprocessor maintains a series connection or a parallel connection betweentwo elements, e.g., a capacitor and an inductor, a resistor and aninductor, a capacitor and a resistor, etc., of the RF transmission model106 before and after the segmentation. As another example, the processormaintains a series connection or a parallel connection between twoelements, e.g., a capacitor and an inductor, a resistor and an inductor,a capacitor and a resistor, etc., of the cable model 104A before andafter the segmentation.

The modules 602, 604, and 606 are coupled with each other. For example,the module 602 is coupled to the module 604 via a link 608 and themodule 606 is coupled to the module 604 via a link 610.

In embodiments in which the cable model 600 is generated by convertingthe cable model 104A, 104B, or 104C, the module 602 has an input 612,which is an example of the input 105A, the input 105B, or the input 105C(FIG. 1). Moreover, the module 602 has an output 614, which is coupledto an input 616 of the module 604. Also, the module 604 has an output618, which is coupled to an input 620 of the module 606. The module 606has an output 622, which is an example of the input 146A, 146B, or 146C(FIG. 1) of the impedance matching model 102.

It should be noted that in various embodiments, when an output of acable model is coupled to an input of the impedance matching model 102,the output is represented by the input and vice versa. For example, anoutput of the cable model 104A is represented by the input 146A, anoutput of the cable model 104B is represented by the input 146B, and anoutput of the cable model 104C is represented by the input 146C.

In embodiments in which the RF transmission model 600 is generated byconverting the RF transmission model 106 (FIG. 1), the input 612 is anexample of the output 148 (FIG. 1) of the impedance matching model 102.It should be noted that in some embodiments, the output 148 is coupledto an input of the RF transmission model 106. In these embodiments, whenthe output 148 is coupled to the input of the RF transmission model 106,the output 148 also represents the input of the RF transmission model106. Moreover, the output 622 is an example of the output 150 (FIG. 1)of the RF transmission model 106.

In various embodiments, the cable model/RF transmission model 600includes modules 602, 604, and 606 per unit length of an RF cable, e.g.,the RF cable 124A, or the RF cable 124B, or the RF cable 124C, etc., orper unit length of the RF transmission line 128 (FIG. 1). For example,the processor 142 (FIG. 1) generates one module of the cable model/RFtransmission model 600 per unit length of the RF cable 124A, or per unitlength of the RF cable 124B, or per unit length of the RF cable 124C, orper unit length of the RF transmission line 128. As another example,when the processor 142 determines that there are 10 unit lengths of theRF transmission line 128, the processor 142 segments an RF transmissionmodel into ten modules. As another example, when the processor 142determines that there are 12 unit lengths of the RF cable 124A, theprocessor 142 segments a cable model into twelve modules.

In some embodiments, the processor 142 determines the unit length of anRF cable or an RF transmission line to be less than a fraction of awavelength of an RF signal that is transferred via the RF cable or theRF transmission line. For example, the unit length is less than 0.1 of awavelength of an RF signal that is transferred via the RF cable or theRF transmission line. As another example, the unit length is less than afraction of a wavelength of an RF signal that is transferred via the RFcable or the RF transmission line, where the fraction ranges from 0.1 to0.2.

FIG. 9 is a diagram of an embodiment of a module d/e, e.g., module d ormodule e, etc., of the RF cable model/Transmission line model 600 (FIG.8), where d ranges from 1 thru D and e ranges from 1 thru E. The moduled/e includes a series circuit 702 and a shunt circuit 704. In someembodiments, the module d/e includes only one series circuit 702 andonly one shunt circuit 704. The shunt circuit 704 is coupled to a groundconnection 707.

The module d/e has an input 706, which is an example of the input 612,the input 616, or the input 620 (FIG. 8). Moreover, the module d/e hasan output 708, which is an example of the output 614, the output 618, orthe output 622 (FIG. 8).

As shown, the series circuit 702 is coupled to the input 706 and to theoutput 708. Moreover, the shunt circuit 704 is coupled to the output708.

In various embodiments, the series circuit 702 or the shunt circuit 704includes a resistor coupled in series with an inductor and a capacitor.In some embodiments, the series circuit 702 or the shunt circuit 704includes a resistor coupled in series with an inductor or in series witha capacitor. In several embodiments, the series circuit 702 or the shuntcircuit 704 includes an inductor coupled in series with a capacitor. Inseveral embodiments, the series circuit 702 or the shunt circuit 704includes an inductor, a resistor, or a capacitor.

In some embodiments, an impedance function is used instead of the seriescircuit 702 and an impedance function is used instead of the shuntcircuit 704. The impedance function that is used instead of the seriescircuit 702 represents a directional sum of impedances of all elementsof the series circuit 702. For example, the series circuit 702 isrepresented as R_(SX)+jX_(SX), where R_(SX) is a result of a directionalsum of resistances of all elements of the series circuit 702, and X_(SX)is a result of a directional sum of reactances of all elements of theseries circuit 702. Moreover, the impedance function that is usedinstead of the shunt circuit 704 represents a directional sum ofresistances of all elements of the shunt circuit 704 and a directionalsum of reactances of all elements of the shunt circuit 704. For example,the shunt circuit 704 is represented as R_(px)+jX_(px), where R_(px) isa result of a directional sum of resistances of all elements of theshunt circuit 704, and X_(px) is a result of a directional sum ofreactances of all elements of the shunt circuit 704.

In some embodiments, the processor 142 (FIG. 1) determines an impedanceZ_((f+1)-in) at an input, e.g., as seen from an input, etc., of an(f+1)^(th) module of the RF cable model/Transmission line model 600based on an impedance Z_(f-in) at an input, e.g., as seen from an input,etc., of an f^(th) module of the RF cable model/Transmission line model600 and parameters of the f^(th) module, where f is d or e. For example,the processor 142 determines the impedance Z_((f+1)-in) according to afunction:

$\begin{matrix}{Z_{{({f + 1})} - {i\; n}} = {R_{0}\frac{Z_{f - {i\; n}} + {R_{0}{\tan( {\beta\; l} )}}}{R_{0} + {Z_{f - {i\; n}}{\tan( {\beta\; l} )}}}}} & (7)\end{matrix}$

where 1 is a length of the corresponding RF transmission line 128, theRF cable 124A, the RF cable 124B, or the RF cable 124C for which theimpedance Z_((f+1)-in) is calculated, R₀ and β are properties of the RFtransmission line 128, the RF cable 124A, the RF cable 124B, or the RFcable 124C. For example, R₀ is the characteristic resistance of the RFtransmission line 128, the RF cable 124A, the RF cable 124B, or the RFcable 124C. The processor 142 determines the property R₀ as being equalto a function:

$\begin{matrix}{R_{0} = \sqrt{\frac{j\;\omega\; L}{j\;\omega\; C}}} & (8)\end{matrix}$

where ω is equal to 2π*frequency, where frequency is a frequency of anRF generator, “L” is an inductance of the series circuit 702 (FIG. 9)and “C” is a capacitance of the shunt circuit 704. The processor 142determines the parameter β as being equal to a ratio of a product of 2and π over a wavelength λ of an RF signal transferring via the RFtransmission line 128, the RF cable 124A, the RF cable 124B, or the RFcable 124C. In some embodiments, a voltage and current probe (not shown)is coupled to the RF transmission line 128, the RF cable 124A, the RFcable 124B, or the RF cable 124C to provide a complex voltage andcurrent of an RF signal that transfers via the RF transmission line 128,the RF cable 124A, the RF cable 124B, or the RF cable 124C to theprocessor 142 and the processor 142 determines the wavelength from thecomplex voltage and current. The processor 142 is coupled to the voltageand current probe (not shown).

In some embodiments, the processor 142 calculates the impedance Z_(f-in)at the input 612 (FIG. 8) based on a complex voltage and currentreceived via the processor 142 from the probe 108 (FIG. 1). For example,when the f^(th) module is a first module of the RF cable model 600, theprocessor 142 calculates the impedance Z_(f-in) at the input 612 as aratio of the complex voltage received via the processor 142 from theprobe 108 and the complex current received via the processor 142 fromthe probe 108.

In various embodiments, the processor 142 calculates the impedanceZ_(f-in) at the input 612 (FIG. 8) based on a complex voltage andcurrent received via the processor 142 from the probe 108 (FIG. 1). Forexample, when the f^(th) module is a first module of the RF transmissionmodel 600, the processor 142 calculates the impedance Z_(f-in) at theinput 612 as a ratio of a complex voltage determined at the input 612and a complex current determined at the input 612. The complex voltageat the input 612 is determined by the processor 142 as a directed sum ofa complex voltage received from the probe 108, a complex voltagedetermined from characteristics of the cable model 104A, and a complexvoltage determined from characteristics of the impedance matching model102 (FIG. 1). The complex current at the input 612 is determined by theprocessor 142 as a directed sum of a complex current received from theprobe 108, a complex current determined from characteristics of thecable model 104A, and a complex current determined from characteristicsof the impedance matching model 102 (FIG. 1).

To generate a cable model/RF transmission model of another RF cable/RFtransmission line (not shown), e.g., a circuit that replaces and isother than that of RF cable 124A, or a circuit that replaces and isother than that of RF cable 124B, or a circuit that replaces and isother than that of RF cable 124C, or a circuit that replaces and isother than that of RF transmission line 128 (FIG. 1), etc., theprocessor 142 replaces the module 602 with another module (not shown),replaces the module 604 with another module (not shown), and/or replacesthe module 606 with another module (not shown). The processor 142establishes a series link between the replacement modules and unreplacedmodules, e.g., the module 602, 604, or 606, etc., when all of themodules 602, 604 and 606 are not replaced or establishes a series linkbetween the replacement modules when all of the modules 602, 604 and 606are replaced by the replacement modules.

A series combination of the other modules (not shown) that replaces thecorresponding modules 602, 604, and/or 606 has similar characteristicsas that of the other replacement RF cable/RF transmission line (notshown). For example, a combined impedance of the other modules (notshown) is the same as or within a range of an impedance of the other RFcable/RF transmission line (not shown). In this example, the othermodules (not shown) represent the other RF cable/RF transmission line(not shown). As another example, a combined impedance of one of theother replacement modules (not shown), the module 604, and the module606 is the same as or within a range of an impedance of the other RFcable/RF transmission line (not shown). In this example, the one of theother replacement modules (not shown), the module 604, and the module606 represent the other RF cable/RF transmission line (not shown).Modularity of RF cable/RF transmission lines allows easy replacement ofone or more modules of one of the RF cable/RF transmission lines withone or more modules of another one of the RF cable/RF transmissionlines.

Upon replacing the module 602 with another module (not shown), replacingthe module 604 with another module (not shown), and/or replacing themodule 606 with another module, the processor 142 checks whethercharacteristics, e.g., impedance, complex voltage and current, etc., ofa cable model/RF transmission model that includes one or more of thereplacement modules (not shown) and/or one or more of the modules 602,604, and 606 are similar to characteristics, e.g., impedance, complexvoltage and current, etc., of the other RF cable/RF transmission line(not shown). For example, the processor 142 calculates a combinedimpedance of the replacement modules (not shown) and/or one or more ofthe modules 602, 604, and 606 and compares the combined impedance withan impedance of the other replacement RF cable/RF transmission line (notshown). Upon determining that the combined impedance of the replacementmodules (not shown) and/or one or more of the modules 602, 604, and 606matches with or is within a range of the impedance of the otherreplacement RF cable/RF transmission line (not shown), the processor 142determines that characteristics of the cable model/RF transmission modelthat includes one or more of the replacement modules (not shown) and/orone or more of the modules 602, 604, and 606 are similar tocharacteristics of the other RF cable/RF transmission line (not shown).On the other hand, upon determining that the combined impedance of thereplacement modules (not shown) and/or one or more of the modules 602,604, and 606 do not match with or are not within a range of theimpedance of the other replacement RF cable/RF transmission line (notshown), the processor 142 determines that characteristics of the cablemodel/RF transmission model that includes one or more of the replacementmodules (not shown) and/or one or more of the modules 602, 604, and 606are not similar to characteristics of the other RF cable/RF transmissionline (not shown).

In various embodiments, the impedance of the other replacement RFcable/RF transmission line is received by the processor 142 from anotherprocessor. In some embodiments, the impedance of the other replacementRF cable/RF transmission line is calculated by the processor 142 basedon complex voltages and currents measured at an input and at an outputof the other replacement RF cable/RF transmission line.

In some embodiments, the series circuit 702 is placed to the right ofthe shunt circuit 704. For example, the series circuit 702 is coupled tothe input 706, the shunt circuit 704, and to the output 708. Moreover,the shunt circuit 704 is coupled to the input 706 and to the groundconnection 706. As another example, the shunt circuit 704 shunts asignal that is received as an input by the series circuit 702.Comparatively, the shunt circuit 704 of the module d/e shunts a signalthat is provided as an output by the series circuit 702. Theseembodiments are similar to the embodiments of the module 229 illustratedwith respect to FIG. 6.

FIG. 10A is a diagram of an embodiment of a module 802, which is anexample of the module d/e (FIG. 9). The module 802 includes a seriesinductor circuit 804 and a parallel capacitor circuit 806. The seriesinductor circuit 804 is an example of the series circuit 702 (FIG. 9)and the parallel capacitor circuit 806 is an example of the shuntcircuit 704 (FIG. 9).

The series inductor circuit 804 includes an inductor L_(cs). Theparallel capacitor circuit 806 includes a capacitor C_(cp). Thecapacitor C_(cp) is coupled to a ground connection 808.

Values of the inductor Lcs and the capacitor C_(cp) are fixed.

FIG. 10B is a diagram of an embodiment of a module 810 in which aninductance of an inductor L_(ms) is variable. The module 810 is anexample of the module d/e (FIG. 9). The module 810 includes a seriesinductor circuit 812 and the parallel capacitor circuit 806. The seriesinductor circuit 812 is an example of the series circuit 702 (FIG. 10A).The series inductor circuit 812 includes the variable inductor L_(ms).The module 810 is the same as the module 802 (FIG. 10A) except that inthe module 810, the fixed inductor L_(cs) is replaced with the variableinductor L_(ms).

FIG. 10C is a diagram of an embodiment of a module 816 in which acapacitance of a capacitor C_(mp) is variable. The module 816 is anexample of the module d/e (FIG. 9). The module 816 includes the seriesinductor circuit 804 and a parallel capacitor circuit 820. The parallelcapacitor circuit 820 is an example of the shunt circuit 704 (FIG. 9).The parallel capacitor circuit 820 includes the variable capacitorC_(mp). The module 816 is the same as the module 802 (FIG. 10A) exceptthat in the module 816, the fixed capacitor C_(cp) is replaced with thevariable capacitor C_(mp).

FIG. 10D is a diagram of an embodiment of a module 822 in which aninductance of the inductor L_(ms) and a capacitance of the capacitorC_(mp) are variable. The module 822 is an example of the module d/e(FIG. 9). The module 822 includes the series inductor circuit 812 andthe parallel capacitor circuit 820. The module 822 is the same as themodule 802 (FIG. 10A) except that in the module 822, the fixed inductorL_(cs) is replaced with the variable inductor L_(ms) and the fixedcapacitor C_(cp) is replaced with the variable capacitor C_(mp).

In some embodiments, a value of inductance of the inductor L_(cs) iszero and/or a value of capacitance of the C_(cp) is zero. In variousembodiments, a value of inductance of the inductor L_(ms) is zero and/ora value of capacitance of the C_(mp) is zero.

FIG. 10E is a diagram of an embodiment of a module 824 that represents afunction 826 applied by the series circuit 702 (FIG. 9) and a function828 that is applied by the shunt circuit 704 (FIG. 9). The function 826is the mathematical function R_(SX)+jX_(SX) and the function 828 is themathematical function R_(px)+jX_(px). The function 828 is a shuntfunction that shunts a current output by the function 826.

In the embodiments described with reference to FIG. 10E, the groundconnection 707 (FIG. 9) is referred to as a ground function.

FIG. 11A is an embodiment of a graph 850 that illustrates a linearrelationship between a voltage measured at an output of an impedancematching circuit and a modeled voltage at an output of a correspondingsegmented impedance matching model. For example, a voltage and currentprobe is coupled to the output of an impedance matching circuit tomeasure a voltage at the output. The modeled voltage is plotted along anx-axis and the measured voltage is plotted along a y-axis. The modeledvoltage may be the voltage V_(n-out). As shown, there is a linearrelationship between the modeled voltage and the measured voltage.Moreover, in some embodiments, the linear relationship in the graph 850is achieved after the processor 142 (FIG. 1) modifies values of aresistor, an inductor, and/or a capacitor in the series circuit 218(FIG. 3) and/or after the processor 142 modifies values of a resistor,an inductor, and/or a capacitor in the shunt circuit 220 (FIG. 3).

FIG. 11B is an embodiment of a graph 852 that illustrates a linearrelationship between a current measured at an output of an impedancematching circuit and a modeled current at an output of a correspondingsegmented impedance matching model. For example, a voltage and currentprobe is coupled to the output of an impedance matching circuit tomeasure a current at the output. The modeled current is plotted along anx-axis and the measured current is plotted along a y-axis. The modeledcurrent may be the current I_(n-out). As shown, there is a linearrelationship between the modeled current and the measured current.Moreover, in some embodiments, the linear relationship in the graph 852is achieved after the processor 142 (FIG. 1) modifies values of aresistor, an inductor, and/or a capacitor in the series circuit 218(FIG. 3) and/or after the processor 142 modifies values of a resistor,an inductor, and/or a capacitor in the shunt circuit 220 (FIG. 3).

FIG. 12A is an embodiment of a graph 854 that illustrates a relationshipbetween a voltage measured at an output of an impedance matching circuitwith respect to time and a modeled voltage at an output of an impedancematching model that is generated based on the impedance matching circuitwith respect to time. The measured voltage and the modeled voltage areplotted along a y-axis and time is plotted on an x-axis. As shown, themodeled voltage overlaps with the measured voltage.

FIG. 12B is an embodiment of a graph 856 that illustrates a relationshipbetween a current measured at an output of an impedance matching circuitwith respect to time and a modeled current at an output of an impedancematching model that is generated based on the impedance matching circuitwith respect to time. The measured current and the modeled current areplotted along a y-axis and time is plotted on an x-axis. As shown, themodeled current overlaps with the measured current.

When one of the x, y, and z MHz RF generators is on, e.g., powered on,etc., and the remaining of the x, y, and z MHz RF generators are off,the processor 142 applies a projected complex voltage and currentdetermined at the output 150 (FIG. 1) as an input to a function to mapthe projected complex voltage and current to a wafer bias value at theoutput 150. For example, when the x, y, or z MHz RF generator is on, awafer bias at the output 150 is determined as a sum of a first producta1*V, a second product b1*I, a third product c1*sqrt(P), and a constantd1, where “sqrt” is square root, V is a voltage magnitude of theprojected complex voltage and current at the output 150, I is a currentmagnitude of the projected complex voltage and current at the output150, P is a power magnitude of the projected complex voltage and currentat the output 150, a1, b1, and c1 are coefficients, and d1 is aconstant. The processor 142 determines the projected complex voltage andcurrent at the output 150 when the x, y, or z MHz RF generator is onbased on the complex voltage and current received at the correspondinginput 105A, 105B, or 105C from a corresponding voltage and current probethat is coupled to the x, y, or z MHz RF generator, an impedance of thecorresponding cable model 600 (FIG. 8) that receives the complex voltageand current from the corresponding voltage and current probe, animpedance of the impedance matching model 103 (FIG. 2), and an impedanceof the RF transmission model 600 (FIG. 8).

Moreover, when two of the x, y, and z MHz RF generators are on and theremaining of the x, y, and z MHz RF generators are off, the processor142 calculates a wafer bias at the output 150 as a sum of a firstproduct a12*V1, a second product b12*I1, a third product c12*sqrt(P1), afourth product d12*V2, a fifth product e12*I2, a sixth productf12*sqrt(P2), and a constant g12, where V1 is a voltage magnitude at theoutput 150 as a result of a first one of the two RF generators being on,I1 is a current magnitude at the output 150 as a result of the first RFgenerator being on, P1 is a power magnitude at the output 150 as aresult of the first RF generator being on, V2 is a voltage magnitude atthe output 150 as a result of a second one of the two RF generatorsbeing on, I2 is a current magnitude at the output 150 as a result of thesecond RF generator being on, and P2 is a power magnitude at the output150 as a result of the second RF generator being on, a12, b12, c12, d12,e12, and f12 are coefficients, and g12 is a constant.

As yet another example, when all of the x, y, and z MHz RF generatorsare on, the processor 142 calculates a wafer bias at the output 150 as asum of a first product a123*V1, a second product b123*I1, a thirdproduct c123*sqrt(P1), a fourth product d123*V2, a fifth producte123*I2, a sixth product f123*sqrt(P2), a seventh product g123*V3, aneighth product h123*I3, a ninth product i123*sqrt(P3), and a constantj123, where V1, I1, P1, V2, I2, and P2 are described above in thepreceding example, V3 is a voltage magnitude at the output 150 as aresult of a third one of the RF generators being on, I3 is a currentmagnitude at the output 150 as a result of the third RF generator beingon, and P3 is a power magnitude at the output 150 as a result of thethird RF generator being on, a123, b123, c123, d123, e123, f123, g123,h123, and i123 are coefficients and j123 is a constant.

In some embodiments, a function used to determine a wafer bias is a sumof characterized values and a constant. The characterized values includemagnitudes, e.g., the magnitudes V, I, P, V1, I1, P1, V2, I2, P2, V3,I3, P3, etc. The characterized values also include coefficients, e.g.,the coefficients, a1, b1, c1, a12, b12, c12, d12, e12, f12, a123, b123,c123, d123, e123, f123, g123, h123, i123, etc. Examples of the constantinclude the constant d1, the constant g12, the constant j123, etc.

It should be noted that the coefficients of the characterized values andthe constant of the characterized values incorporate empirical modelingdata. For example, wafer bias is measured for multiple times within theplasma chamber 130 (FIG. 1) by using a wafer bias sensor. Moreover, inthe example, for the number of times the wafer bias is measured, complexvoltages and currents at output 150 are determined by the processor 142based on the complex voltage and current from one or more of theoutputs, e.g., the output 110, 114, 118 (FIG. 1), etc., of one or moreof the RF generators, e.g., the x MHz RF generator, the y MHz RFgenerator, the z MHz RF generator, etc., based on an impedance of thecable model 600 (FIG. 8), an impedance of the impedance matching model103 (FIG. 2), and an impedance of the RF transmission model 600 (FIG.8). Moreover, in this example, a statistical method, e.g., partial leastsquares, best fit, fit, regression, etc., is applied by the processor142 to the measured wafer bias and to voltage magnitudes, currentmagnitudes, and power magnitudes extracted from the complex voltages andcurrents at the output 150 to determine the coefficients of thecharacterized values and the constant of the characterized values.

In some embodiments, a function used to determine a wafer bias is apolynomial.

It is noted that although the above-described operations are describedwith reference to a parallel plate plasma chamber, e.g., a capacitivelycoupled plasma chamber, etc., in some embodiments, the above-describedoperations apply to other types of plasma chambers, e.g., a plasmachamber including an inductively coupled plasma (ICP) reactor, atransformer coupled plasma (TCP) reactor, conductor tools, dielectrictools, a plasma chamber including an electron-cyclotron resonance (ECR)reactor, etc. For example, the x MHz RF generator, the y MHz RFgenerator, and the z MHz RF generator are coupled to an inductor withinthe ICP plasma chamber.

It is also noted that although the operations above are described asbeing performed by the processor 142 (FIG. 1), in some embodiments, theoperations may be performed by one or more processors of the host system143 or by multiple processors of multiple host systems.

It should be noted that although the above-described embodiments relateto providing an RF signal to the lower electrode of the ESC 132 (FIG. 1)and grounding the upper electrode 134 (FIG. 1), in several embodiments,the RF signal is provided to the upper electrode 134 while the lowerelectrode of the ESC 132 is grounded.

Embodiments described herein may be practiced with various computersystem configurations including hand-held hardware units, microprocessorsystems, microprocessor-based or programmable consumer electronics,minicomputers, mainframe computers and the like. The embodiments canalso be practiced in distributed computing environments where tasks areperformed by remote processing hardware units that are linked through anetwork.

With the above embodiments in mind, it should be understood that theembodiments can employ various computer-implemented operations involvingdata stored in computer systems. These operations are those requiringphysical manipulation of physical quantities. Any of the operationsdescribed herein that form part of the embodiments are useful machineoperations. The embodiments also relates to a hardware unit or anapparatus for performing these operations. The apparatus may bespecially constructed for a special purpose computer. When defined as aspecial purpose computer, the computer can also perform otherprocessing, program execution or routines that are not part of thespecial purpose, while still being capable of operating for the specialpurpose. In some embodiments, the operations may be processed by ageneral purpose computer selectively activated or configured by one ormore computer programs stored in the computer memory, cache, or obtainedover a network. When data is obtained over a network, the data may beprocessed by other computers on the network, e.g., a cloud of computingresources.

One or more embodiments can also be fabricated as computer-readable codeon a non-transitory computer-readable medium. The non-transitorycomputer-readable medium is any data storage hardware unit that canstore data, which can be thereafter be read by a computer system.Examples of the non-transitory computer-readable medium include harddrives, network attached storage (NAS), ROM, RAM, compact disc-ROMs(CD-ROMs), CD-recordables (CD-Rs), CD-rewritables (CD-RWs), magnetictapes and other optical and non-optical data storage hardware units. Thenon-transitory computer-readable medium can include computer-readabletangible medium distributed over a network-coupled computer system sothat the computer-readable code is stored and executed in a distributedfashion.

One or more features from any embodiment may be combined with one ormore features of any other embodiment without departing from the scopedescribed in various embodiments described in the present disclosure.

Although the foregoing embodiments have been described in some detailfor purposes of clarity of understanding, it will be apparent thatcertain changes and modifications can be practiced within the scope ofthe appended claims. Accordingly, the present embodiments are to beconsidered as illustrative and not restrictive, and the embodiments arenot to be limited to the details given herein, but may be modifiedwithin the scope and equivalents of the appended claims.

The invention claimed is:
 1. A system for segmenting an impedancematching model, the method comprising: a radio frequency (RF) generatorconfigured to generate an RF signal; an impedance matching circuitcoupled to the RF generator via an RF cable, wherein the impedancematching circuit is configured to receive the RF signal and generate amodified RF signal; a plasma chamber coupled to the impedance matchingcircuit via an RF transmission line to receive the modified RF signal;and a host computer in communication with the RF generator, wherein thehost computer is configured to: generate the impedance matching modelrepresenting the impedance matching circuit, wherein the impedancematching model includes a first module for a portion of the impedancematching circuit; and replace the first module with one or more othermodules when the impedance matching circuit is replaced with anotherimpedance matching circuit.
 2. The system of claim 1, wherein the firstmodule includes a series circuit.
 3. The system of claim 2, wherein theseries circuit includes a combination of a resistor, a capacitor, and aninductor.
 4. The system of claim 2, wherein the first module is coupledto a second module, wherein the second module is coupled between thefirst module and a computer-generated model of the RF cable, wherein theseries circuit has a first end that is coupled to the second module,wherein the series circuit has a second end that is coupled to acomputer-generated model of the RF transmission line.
 5. The system ofclaim 2, wherein the first module is coupled to a second module, whereinthe second module is located between the first module and acomputer-generated model of the RF transmission line, wherein the seriescircuit has a first end that is coupled to a computer-generated model ofthe RF cable and has a second end that is coupled to the second module.6. The system of claim 1, wherein the first module includes a shuntcircuit having a first end that is coupled to a ground connection. 7.The system of claim 6, wherein the shunt circuit includes a combinationof a resistor, a capacitor, and an inductor.
 8. The system of claim 6,wherein the first module is coupled to a second module, wherein thesecond module is coupled between the first module and acomputer-generated model of the RF cable, wherein the shunt circuit hasa second end that is coupled to the second module and to acomputer-generated model of the RF transmission line.
 9. The system ofclaim 6, wherein the first module is coupled to a second module, whereinthe second module is coupled between the first module and acomputer-generated model of the RF transmission line, wherein the shuntcircuit has a second end that is coupled to the second module and to acomputer-generated model of the RF cable.
 10. The system of claim 1,wherein the first module is a polynomial function defining a seriescircuit.
 11. The system of claim 10, wherein the polynomial functionincludes a combination of a resistance and a reactance.
 12. The systemof claim 1, wherein the first module is a polynomial function defining ashunt circuit.
 13. The system of claim 12, wherein the polynomialfunction includes a combination of a resistance and a reactance.
 14. Thesystem of claim 1, wherein the one or more other modules represent aportion of another impedance matching circuit.
 15. A host computer forsegmenting an impedance matching model, comprising: a processorconfigured to: generate the impedance matching model, the impedancematching model representing an impedance matching circuit, the impedancematching circuit configured to be coupled to a radio frequency (RF)generator via an RF cable and to a plasma chamber via an RF transmissionline, wherein the impedance matching model includes a first module for aportion of the impedance matching circuit; and replace the first modulewith one or more other modules when the impedance matching circuit isreplaced with another impedance matching circuit; and a memory devicecoupled to the processor for storing the impedance matching model. 16.The host computer of claim 15, wherein the first module is a function ora circuit.
 17. The host computer of claim 15, wherein the first moduleincludes a series circuit.
 18. The host computer of claim 17, whereinthe series circuit includes a combination of a resistor, a capacitor,and an inductor.
 19. The host computer of claim 17, wherein the firstmodule is coupled to a second module, wherein the second module iscoupled between the first module and a computer-generated model of theRF cable, wherein the series circuit has a first end that is coupled tothe second module, wherein the series circuit has a second end that iscoupled to a computer-generated model of the RF transmission line. 20.The host computer of claim 17, wherein the first module is coupled to asecond module, wherein the second module is located between the firstmodule and a computer-generated model of the RF transmission line,wherein the series circuit has a first end that is coupled to acomputer-generated model of the RF cable and has a second end that iscoupled to the second module.
 21. The host computer of claim 15, whereinthe first module includes a shunt circuit having a first end that iscoupled to a ground connection.
 22. The host computer of claim 21,wherein the shunt circuit includes a combination of a resistor, acapacitor, and an inductor.
 23. The host computer of claim 21, whereinthe first module is coupled to a second module, wherein the secondmodule is coupled between the first module and a computer-generatedmodel of the RF cable, wherein the shunt circuit has a second end thatis coupled to the second module and to a computer-generated model of theRF transmission line.
 24. The host computer of claim 21, wherein thefirst module is coupled to a second module, wherein the second module iscoupled between the first module and a computer-generated model of theRF transmission line, wherein the shunt circuit has a second end that iscoupled to the second module and to a computer-generated model of the RFcable.
 25. The host computer of claim 15, wherein the first module is apolynomial function defining a series circuit.
 26. The host computer ofclaim 25, wherein the polynomial function includes a combination of aresistance and a reactance.
 27. The host computer of claim 15, whereinthe first module is a polynomial function defining a shunt circuit. 28.The host computer of claim 27, wherein the polynomial function includesa combination of a resistance and a reactance.
 29. The host computer ofclaim 15, wherein the one or more other modules represent a portion ofthe other impedance matching circuit.